Dense Associative Memory on S¹: Phase-Gate Computing and Superlinear Capacity in Circular Oscillator Networks

Dense Associative Memory on S1 — v2.0: Peer-Review Edition We present Dense Associative Memory (DAM) extended to the unit circle S1, where each neuron carries a phase in [0,2pi) rather than a binary spin. The energy function E = -sum_mu F(sum_i cos(phi_i - xi_i^mu)) generalizes the Krotov-Hopfield Dense AM from {+-1}^N to S1N. We prove fixed-point stability analytically and show empirically that F=exp achieves storage capacity alpha*=1.0 for N in {32, 64, 128} -- a 7.2-fold improvement over classical Hopfield (alpha*=0.138). The F=exp update is formally equivalent to Transformer self-attention with circular inner products. The same dynamics implement universal Boolean gates at 100% accuracy. Physical substrate: 200 Hz-anchored phase oscillator arrays (REZON architecture). DOI: 10.5281/zenodo.18800042 WHAT'S NEW IN THIS VERSION Paper: - New Related Work section (rotor Hopfield, Modern Hopfield, Kuramoto 2025, NeurIPS 2024) - Theorem 1: full 3-step proof, assumption corrected to P < F'(N)/F'(0), covers P=N for F=exp - Euler instability derived from Lipschitz bound: dt*3N^2 < 2 (not a heuristic) - Explicit Hamming definition: b_i = 1[cos(phi_i) < 0] - Full protocol: omega_i ~ N(0,0.01) rad/s, jitter sigma=0.05 rad, bit-flip+Gaussian - N=128 results throughout: capacity tables, full sweep, capacity curve figure - New subsection: role of 200 Hz anchor (reference frame / hardware enabler / perturbation) - Turing completeness qualified: functional completeness empirically verified; Turing completeness under unbounded-memory assumptions - 7.2-fold claim qualified: same metric alpha*, across different model families Experiments: - N=128 full sweep (8.3h, Jetson Orin NX): F=exp alpha*=1.000 (384/384 trials), F=x^2 P*=20 alpha*=0.156 - Phase Attention: Hamming 12->0 in 1 step at N=128, P=5 - CNOT: 100% pass rate, 20 seeds, Wilson 95% CI [0.83, 1.00] Tests: 66 tests passing, GitHub Actions CI green WHAT THE FULL PROJECT CONTAINS phase_dense_am.py -- Dense AM: F=exp/poly2/poly3/linear, circular attention, capacity sweep phase_gate_universal.py -- NOT, AND, OR, XOR, NAND, NOR, half-adder via injection-locking ODE cnot_phase_gate.py -- CNOT with Wilson CI, noise sweep phase_dlatch.py -- Bistable phase latch, robustness tests phase_automaton.py -- 3-state FSM as phase attractors phase_full_adder.py -- 1-bit adder with carry phase_turing_demo.py -- Cascaded memory + logic + sequential computation phase_neural_net.py -- PhaseNN: ODE-based classifier, 664 params vs MLP 5768 phase_hopfield.py -- Baseline: validates alpha*=0.138 paper.tex / paper.pdf -- Full scientific paper, 633 KB test_*.py (8 files) -- 66 tests, pytest CI reports/*.json -- N=32/64/128 capacity, CNOT, FSM results REPRODUCIBILITY.md -- 8-step protocol from scratch to results FORMAL_APPENDIX.md -- Lemmas A1-D2, full proofs THREATS_TO_VALIDITY.md -- Threat analysis for reviewers Code: https://github.com/krisss0mecom/REZON DOI: https://doi.org/10.5281/zenodo.18800042